Locally Decodable Codes
Sergey YekhaninMicrosoft Research
Data storage
• Store data reliably• Keep it readily available for
users
Data storage: Replication
• Store data reliably• Keep it readily available for
users
• Very large overhead• Moderate reliability
• Local recovery: Loose one machine, access one
Data storage: Erasure coding
• Store data reliably• Keep it readily available for
users
• Low overhead• High reliability
• No local recovery: Loose one machine, access k
…
……
k data chunks n-k parity chunks
Need: Erasure codes with local decoding
Local decoding: example
X1
X
E(X)
• Tolerates 3 erasures• After 3 erasures, any information bit can recovered with locality 2• After 3 erasures, any parity bit can be recovered with locality 2
X2 X3
X1
X1X2
X2 X3
X1X3
X2X3
X1X2X3
Local decoding: example
X1
X
E(X)
• Tolerates 3 erasures• After 3 erasures, any information bit can recovered with locality 2• After 3 erasures, any parity bit can be recovered with locality 2
X2 X3
X1
X1X2
X2 X3
X1X3
X2X3
X1X2X3
Locally Decodable Codes
Definition : A code is called - locally decodable if for all and all values of the symbol can be recovered from accessing only symbols of , even after an arbitrary 10% of coordinates of are erased.
0 0 1 0 1 … 0 1 1
0 1 … 0 1
0 1 0 … 0 1
k long message
n long codewordAdversarial
erasures
Decoder reads only r symbols
nq
kq FFC : r
kqFx ],[ki ix
r )(xC)(xC
ParametersIdeally:
– High rate: close to . or
– Strong locality: Very small Constant.
One cannot minimize and simultaneously. There is a trade-off.
n .)1( kn
r
k
.r
)(kOn
n
Parameters
Ideally:– High rate: close to . or
– Strong locality: Very small Constant.
n k )(kOn .)1( kn
.rApplications in complexity theory / cryptography.
Potential applications for data transmission / storage.
Early constructions: Reed Muller codes
Parameters:The code consists of evaluations of all degree
polynomials in variables over a finite field
• High rate: No locality at rates above 0.5 Locality at rate
• Strong locality: for constant .r
.qFmd
.,, dmq
)1/(1
2
rkn
./1 Okr
State of the art: codes
• High rate: [KSY10]
Multiplicity codes: Locality at rate
• Strong locality: [Y08, R07, KY09,E09, DGY10, BET10a, IS10, CFL+10,BET10b,SY]
Matching vector codes: for constant
for
.r
.1
)(log22k
n
kr
k
nlog22 .3r
State of the art: lower bounds [KT,KdW,W,W]
• High rate: [KSY10]
Multiplicity codes: Locality at rate
• Strong locality: [Y08, R07, E09, DGY10, BET10a, IS10, CFL+10,BET10b,SY11]
Matching vector codes: for constant
for
kr .1
)(log22k
n .r.3r
k
nlog22
)(log kr
)/1(1 rkn Length lower bound:
Locality lower bound:
State of the art: constructions
Matching vector codes Reed Muller codes Multiplicity codes
kr ck
kr)log(log
log
2r kr log2
Plan
• Reed Muller codes
• Multiplicity codes
• Matching vector codes
Reed Muller codes• Parameters: • Code: Evaluations of degree polynomials
over • Set: • Polynomial yields a codeword:
• Parameters:
.,, dmqd .mqF
.)1(,2 qdm
,21
21/
22 2
2
qd
nk .kqr ,2qn
],[ 21 zzFf q
2)(qFx
xf
Reed Muller codes: local decoding• Key observation: Restriction of a codeword to an
affine line yields an evaluation of a univariate polynomial of degree
• To recover the value at – Pick an affine line through with not too many
erasures.– Do polynomial interpolation.2
qF
:x
x
x
.d
• Locally decodable code: Decoder reads random locations.
Lf
k
Multiplicity codes
Multiplicity codes• Parameters: • Code: Evaluations of degree polynomials
over and their partial derivatives.
• Set:• Polynomial yields a codeword:
• Parameters:
.,, dmqd m
qF
2
)(),(),(21
qFx
xzfx
zfxf
.)1(2,2 qdm ],[ 21 zzFf q
.22 kqr ,3/26
143/2
2 22
qd
nk,3 2qn
qd
Multiplicity codes: local decoding
• Fact: Derivatives of in two independent directions determine the derivatives in all directions.
• Key observation: Restriction of a codeword to an affine line yields an evaluation of a univariate polynomial of degree L
f .d
f
Multiplicity codes: local decoding• To recover the value at
– Pick a line through . Reconstruct– Pick another line through . Reconstruct– Polynomials and determine
• Increasing multiplicity yields higher rate. • Increasing the dimension yields smaller query complexity.
2qFx
1Lf
1L2L 2L
f
)(),(),(
21
xzfx
zfxf1L
f2L
f
:x
xx
RM codes vs. Multiplicity codes
Reed Muller codes Multiplicity codes
Codewords
Evaluations of polynomials
Higher order evaluations of polynomials
Evaluation domain
All of the domain All of the domain
Decoding Along a random affine line
Along a collection of random affine lines
Locally correctable
Yes Yes
Matching vector codes
Matching vectors
• Definition: Let
We say that form a matching family if :– For all– For all
• Core theorem: A matching vector family of size yields an query code of length
},,...,{ 1 kuuU ,, hmZVU
).,(0 ij vu
}.,...,{ 1 kvvV
;0),(, ii vuiVU ,
k
,ji
m .hmn
MV codes: Encoding• Let contain a multiplicative subgroup of
size • Given a matching family• A message: • Consider a polynomial in the ring:
• Encoding is the evaluation of over
),...,( 1 kbb
hmZVU ,
],...,[ 1 hq zzF
k
j
uj
jzbF1
F hC
hC
)()2(2
)1(1 ... hu
huuu jjjj zzzz
.mqF C
Multiplicity codes: local decoding• Concept: For a multiplicative line
through in direction
• Key observation: evaluation of is a evaluation of a univariate polynomial whose term determines
• To recover – Pick a multiplicative line – Do polynomial interpolation
phm
h ZvCp ,:v }|{, CxxpM v
vp
ivp
M , CF.ib
hCibivp
M ,
),(
,
ijj
vp
j vuujM
uj xpbzb
RM codes vs. Multiplicity codes
Reed Muller codes Multiplicity codes
Codewords
Evaluations of low degree
polynomials
Evaluations of polynomials with specific monomial
degreesEvaluation domain
All of the domain A subset of the domain
Decoding Along a random affine line
Along a random multiplicative line
Locally correctable
Yes No
Summary
• Despite progress, the true trade-off between codeword length and locality is still a mystery.– Are there codes of positive rate with ?– Are there codes of polynomial length and
?
• A technical question: what is the size of the largest family of subsets of such that– For all modulo six; – For all modulo six.
)(logkOr )1(okr
},...,{ 1 kuu ][n,i
,ji 0|| iu
0|| ji uu
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